By Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres

ISBN-10: 1470409100

ISBN-13: 9781470409104

The Swendsen-Wang dynamics is a Markov chain time-honored through physicists to pattern from the Boltzmann-Gibbs distribution of the Ising version. Cooper, Dyer, Frieze and Rue proved that at the whole graph Kn the blending time of the chain is at such a lot O( O n) for all non-critical temperatures. during this paper the authors express that the blending time is Q (1) in excessive temperatures, Q (log n) in low temperatures and Q (n 1/4) at criticality. in addition they offer an top certain of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts version on any tree of n vertices

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**Extra info for A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics**

**Example text**

Note that |Xt | and the original magnetization chain has the same distribution. 1 gives the required upper bound and concludes the proof. 1. 3. [Crossing and overshoot] Let Xt and Yt be two SW magnetizad tion chains with X0 ≥ n3/4 and Y0 = π. Put T = min t : Xt , Yt ∈ [A−1 n3/4 , An3/4 ] and |Xt − Yt | ≤ hn5/8 , for some constant h > 0 and large constant A. Then we can choose positive constants h, q, K depending only on A such that P(T ≤ Kn1/4 ) ≥ q . 4. [Local CLT] For any constants A > 1 and h > 0, there exist constants δ = δ(A, h) > 0 and k = k(A, h) ∈ N such that for any x0 ∈ [A−1 n3/4 , An3/4 ] and any x ∈ n + 2Z with |x − x0 | ≤ hn5/8 , we have P(Xk = x|X0 = x0 ) ≥ δn−5/8 .

34) is o(1). 34). Since EX = mP(|C(v)| ≥ ) it suﬃces to bound from above P(|C(v)| ≥ ). Recall that |C(v)| is the hitting time of the process {Yt } at 0. 16. Let τ = mint {Wt = 0} be the hitting time of Wt at 0, then it is clear that we can couple Wt and Yt such that |C(v)| ≤ τ . Thus 3 1 20 P(|C(v)| ≥ ) ≤ P(τ ≥ ) = P(τ = ∞) + P( ≤ τ < ∞) . 16 with T = = 20 m and get by the previous display that 3 8 P(|C(v)| ≥ m/20) ≤ 2 − 2 + O( 3 ) + C1 −7/2 m−3/2 e− m/4 3 8 = 2 − 2 + O( 3 ) , 3 as long as 3 m ≥ A log m for large enough A.

8. 18. Let Xt be a magnetization chain with X0 ∈ [b1 n3/4 , b2 n3/4 ] where b2 > b1 > 0 are two constants. Let τ1 be the ﬁrst time that Xt ∈ [ b21 n3/4 , (b2 + b1 3/4 ]. Then there exists a constant C = C(b1 , b2 ) > 0 such that for all constant 2 )n δ > 0 we have P(τ1 ≤ δn1/4 ) ≤ Cδ 2 . 18) E (X(t+1)∧τ1 − Xt∧τ1 )k Ft ≤ Cn5k/8 b1 3/4 ]. 2 )n Part 58 YUN LONG, ASAF NACHMIAS, WEIYANG NING, and YUVAL PERES for k = 2, 3, 4. Deﬁne Z := X(t+1)∧τ1 − Xt∧τ1 − (EX(t+1)∧τ1 − EXt∧τ1 ). 19) 5k 8 for k = 2, 3, 4.

### A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics by Yun Long, Asaf Nachmias, Weiyang Ning, Yuval Peres

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